How long has he had this nickname and when and how did he first get it?

Parcells earned the nickname "The Big Tuna" when he responded to a statement from a player with the incredulous "Who do you think I am? Charlie the Tuna?" Link below).

or

From "The Tuna" himself........(interview).

[Coach Parcells discusses how he got the nickname ''Tuna'' at a 1997 press conference.] Bill Parcells: I think it goes back to my first time with the Patriots. There was an old commercial from StarKist with Charlie the StarKist tuna. So my players were trying to con me on something one time, and I said, 'You must think I'm Charlie the Tuna,' you know, a sucker, and that's kind of how it started. We started with it that year and [Charlie and the other fish] used to wear those little tuna helmets, you know, tuna pictures on their helmets. That's where it all started.

Parcells earned the nickname "The Big Tuna" when he responded to a statement from a player with the incredulous "Who do you think I am? Charlie the Tuna?"

because he likes TUna.lol.

He's a fish.

"There was an old commercial from Starkist with Charlie the Starkist Tuna, so my players were trying to con me on something one time, and I said, 'You must think I'm Charlie the Tuna,' you know, a sucker, and that's kind of how it started.

"We started with it that year and they used to wear those little tuna helmets, you know, tuna pictures on their helmets. That's where it all started."